{
  "id": "1512.08473",
  "version": "v1",
  "published": "2015-12-28T18:10:12.000Z",
  "updated": "2015-12-28T18:10:12.000Z",
  "title": "Shotgun assembly of random regular graphs",
  "authors": [
    "Elchanan Mossel",
    "Nike Sun"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In recent work, Mossel and Ross (2015) consider the shotgun assembly problem for random graphs G: what radius R ensures that G can be uniquely recovered from its list of rooted R-neighborhoods, with high probability? Here we consider this question for random regular graphs of fixed degree d (at least three). A result of Bollobas (1982) implies efficient recovery at $R=((1+\\epsilon)\\log n)/(2\\log(d-1))$ with high probability --- moreover, this recovery algorithm uses only a summary of the distances in each neighborhood. We show that using the full neighborhood structure gives a sharper bound $R = (\\log n + \\log\\log n)/(2\\log(d-1))+ O(1)$, which we prove is tight up to the O(1) term. One consequence of our proof is that if G,H are independent graphs where G follows the random regular law, then with high probability the graphs are non-isomorphic; and this can be efficiently certified by testing the R-neighborhood list of H against the R-neighborhood of a single adversarially chosen vertex of G.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-28T18:10:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C60"
    ],
    "keywords": [
      "random regular graphs",
      "high probability",
      "r-neighborhood",
      "random regular law",
      "full neighborhood structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}